Antenna Array Signal Processing for Multistatic Radar Systems

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Antenna Array Signal Processing for

Multistatic Radar Systems

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Antenna Array Signal Processing for

Multistatic Radar Systems

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof.dr.ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op vrijdag 12 juli 2013 om 10:00 uur

door

Francesco BELFIORI

Laurea Specialistica in Ingegneria delle Telecomunicazioni Universit`a degli studi di Roma “La Sapienza”

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Dit proefschrift is goedgekeurd door de promotor: Prof.ir. P. Hoogeboom

Samenstelling promotiecomissie: Rector Magnificus, voorzitter

Prof. ir. P. Hoogeboom, Technische Universiteit Delft, promotor Prof. Dr. L. Ferro-Famil, Universit´e de Rennes 1

Prof. Dr. A. Yarovoy, Technische Universiteit Delft

Prof. Dr.-Ing. J. Ender, Universit¨at Siegen - Fraunhofer FHR Prof. Dr. F. Le Chevalier, Technische Universiteit Delft - Thales Dr. W. van Rossum, TNO

This research was supported by TNO under contract DenV-017.

ISBN 978-94-6191-782-9

Antenna Array Signal Processing for Multistatic Radar Systems. Dissertation at Delft University of Technology.

Copyright c 2013 by Francesco Belfiori.

All rights reserved. No parts of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the author.

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List of Figures

1.1 Sketch of an analog beamformer based on phase shifter components. . 2 1.2 Basic digital beamforming scheme. . . 3 2.1 Generic antenna array geometry. . . 12 2.2 Linear antenna array geometry scanning on the xOz plane. . . 14 2.3 Normalised array pattern behaviors along the azimuthal plane (θ = 90o_{). 17}

2.4 Circular antenna array geometry. . . 18 2.5 3D view of the array factor for an 8-elements UCA (linear scale). . . . 19 2.6 Array factor for an 8-elements UCA along the azimuthal (θ = 90o, left)

and the elevation (φ = 0o_{, right) planes. . . .} ₁₉

3.1 Digital signal processing scheme of the PCL radar. . . 24 3.2 First kind Bessel functions of different orders. . . 29 3.3 Normalised UCA pattern and mask function fd(φ) of the desired

pat-tern behavior. . . 30 3.4 Results of the proposed side lobe reduction method for a (a) UCA with

radius r = 0.48λ, (b) UCA with radius r = 0.36λ. . . 31 3.5 Results of the phase modes side lobe reduction method for a (a) UCA

with radius r = 0.48λ, (b) UCA with radius r = 0.36λ. . . 32 3.6 Comparison between the phase modes and the proposed DBF

tech-nique syntheses of a circular array pattern with radius r = 0.36λ and SLL=−19dB. . . 33 4.1 Block diagram of the PCL system. . . 38 4.2 TNO circular array for passive radar applications. . . 39 4.3 Simulated element pattern gain of a stand alone dipole considered as a

single radiating element using CST: (a) 3D plot and (b) cut along the elevation plane (φ = 90◦). . . 40

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4.4 Simulated element pattern gain of a single dipole in the circular array configuration using CST: (a) 3D plot and (b) cut along the azimuthal plane (θ = 90◦). . . 40 4.5 Comparison between the measured element patterns for three different

array channels and the CST simulated data. . . 41 4.6 Array construction model used in the CST simulator. . . 42 4.7 Simulated circular array pattern gain using CST: (a) 3D plot and (b)

“blue line”: cut along the azimuthal plane (θ = 90◦) “red line”: cut along the elevation plane (φ = 90◦). . . 42 4.8 Rack of the PCL system analog receiver: (a) front view (b) internal view. 43 4.9 Block diagram of a receiving channel. . . 44 4.10 FM bandwidth input signals measured with the 6-th dipole of the array. 46 4.11 Representation of: a uniformly (a) and a sparsely (b) filled FM band. 47 4.12 (a) Sxx scattering parameters for the 8 channels of the array , (b) S1x

scattering parameters with respect to the first element of the array. . . 48 4.13 (a) Relative (with respect to first array channel) phase shifts after

dig-ital conversion, (b) Measured signal amplitudes after digdig-ital conversion. 48 4.14 Cartesian reference system comparison between the un-/calibrated and

the theoretical patterns in dB scale for (a) transmission point 1 and (b) transmission point 2. . . 49 4.15 Effect of the DBF nulling procedure on the array pattern behavior. . . 50 4.16 Matched filter output Range-Doppler map. . . 52 4.17 GO-CFAR Time vs Range output map (a) and Overlapping of the

CFAR detections with the ADSB available tracks data (b). . . 53 4.18 GO-CFAR Doppler Velocity vs Range output map (a) and Overlapping

of the CFAR detections with the ADSB available tracks data (b). . . . 53 5.1 Periodical array configuration of transmitting and receiving elements. 58 5.2 Different MIMO pattern contributions and realised pattern synthesis. . 61 5.3 Error affected array pattern comparisons for (a) ∆2_{= σ}2

δ = 0.01, (b)

∆2_{= 0.01 and σ}2

δ = 0.1, (c) ∆2= σδ2= 0.1. . . 70

5.4 Analysis of the upper bound conditions for the average error patterns for (a) ∆2_{= σ}2

δ = 0.001, (b) ∆2= 0.01 and σ2δ = 0.1. . . 72

6.1 X-Band radar test board with MIMO functionality developed at TNO. 77 6.2 Details of the USB 2.0 connector (a), boxed test board (b) and radar

system layout while connected to the data acquisition notebook (c). . 77 6.3 Virtual element relative positions of the MIMO array board. . . 78 6.4 Layout of λ/4 (8mm) spaced microstrip fed quasi-Yagi antenna. . . 79 6.5 Scattering parameter measurements of the λ/4 spaced elements. . . . 79

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LIST OF FIGURES xiii

6.6 Required spacings of the receiver elements for increasing number of the transmitters. . . 81 6.7 Measured and expected phase behavior of the reference scatterers. . . 82 6.8 Measured and average (a) phase offset and (b) fractional amplitude

values for the reference scatterers. . . 82 6.9 Effect of the calibration on the pattern synthesis (a) first scatterer (b)

second scatterer. . . 83 6.10 Effect of the tapering on the calibrated and un-calibrated patterns (a)

first scatterer (b) second scatterer. . . 83 6.11 Example of a FMCW transmission scheme for a TDM MIMO array. . 85 6.12 FMCW Range/Doppler processing (a) FMCW Range/Doppler

pro-cessing with a 3 stages MTI canceler (b). . . 86 6.13 TDM MIMO 3D matrix data structure. . . 88 6.14 Ambiguity functions of a target with status vector v0= [50m, 20o, 3m/s]

in a sequential TDM transmission mode: Range/Doppler (a) and An-gle/Doppler (b) maps. . . 89 6.15 Angle/Doppler map for a target with status vector v0= [50m, 20o, 3m/s]

in a conventional FMCW radar with transmitted pulse length of NTT . 90

6.16 Ambiguity functions of a target with status vector v0= [50m, 20o, 3m/s]

in a random TDM transmission mode: range/Doppler (a) and az-imuth/Doppler (b) maps. . . 91 6.17 Ambiguity functions of two targets with status vectors v0= [50m, 20o, 3m/s]

and v1 = [50m, 35o, 10m/s] in a sequential (a) and random (b) TDM

transmission mode. . . 92 6.18 Effect of the number of integrated sweeps on the sidelobes level: (a) 16

MIMO sweeps (b) 128 MIMO sweeps. . . 92 6.19 Data matrix samples and scanning window procedure. . . 96 6.20 Simulated scenario with 6 targets spaced by 0.6m in range and 5o _in

angle: DBF processing (a) 2D-MUSIC processing (b). . . 98 6.21 Cut at the 20.6m range bin for the 3 scatterers case (a); cut at the 20m

range bin for the 2 scatterers case (b). . . 99 6.22 Cut at the 5o _{angular bin for the 3 scatterers case (a); cut at the 0}o

angular bin for the 2 scatterers case (b). . . 99 6.23 Scenario for the measured data set collection with highlighted targets. 100 6.24 Real data scenario: DBF processing (a) 2D-MUSIC processing (b). . . 101 A.1 Set-up of the gain measurements. . . 109 A.2 Set-up of the noise figure measurements. . . 110

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A.3 Element pattern measurement setup: reference transmitter (a), PCL array ARxand reference transmitter AT x(b), measurements geometry

(c) and rotating platform with scaled plane (d) . . . 113 C.1 Transmitted and received sawtooth modulated signals . . . 121

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List of Tables

3.1 Azimuthal resolution performance . . . 32

3.2 Angular resolution comparison for different SLL taperings . . . 32

3.3 Illumination efficiency comparison for different SLL taperings . . . 33

4.1 ICS-554B digitiser board specifications . . . 38

4.2 Average power levels at the input of the analog receiver . . . 45

6.1 Main to Maximum Sidelobe Level . . . 93

6.2 Radar parameters selection for the simulated scenario . . . 98

6.3 Target positions for the simulated scenario . . . 99

A.1 Gain values of the receiver channels . . . 110

A.2 Noise figure values . . . 111

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Chapter 1 Introduction

The basic tasks of a radar system conceive the detection of a target and the esti-mation of its range, i.e. the distance from the radar. This inforesti-mation is retrieved by transmitting a probing signal inside the scene of interest, and by analysing the signal that is reflected by the object. In the case of classical monostatic radars, a single antenna is used at both the transmitting and the receiving sections. When the angular information is needed, the antenna is mechanically rotated and the direction of arrival of the target is obtained.

Modern radars employ transceiver sections that are composed by multiple radiators, which are referred to as array of antennas, in order to enhance the overall system capabilities. With respect to single antenna-based radars, one of the main advantages offered by such systems is the possibility of combining the signals received by the different channels and retrieving the angular information belonging to various direc-tions [1]. This allows for non mechanical scanning of the beams. A higher flexibility for surveillance tasks exploitation can then be achieved. Aside from the targets of in-terest, the radar scenario is characterised by the presence of unwanted signals that can degrade the detection performance of the desired echoes. Those signals can be gener-ated by either environmental reflections (clutter, multipath) or intentional interferers (jammer). The latter class of signals are normally referred to as Electronic Counter Measure (ECM) and they are present in military scenarios with the aim of deceiving the sensor, increasing the noise floor of the radar or saturating the related receiver. Since these disturbances are usually located in well defined sectors and the degrees of freedom provided by the multiple antennas allow the synthesis of directional nulls in the array radiation pattern, several Electronic Counter Counter Measure (ECCM) techniques can be adopted to suppress or mitigate the impact of the interfering signals on the radar system [2].

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Figure 1.1: Sketch of an analog beamformer based on phase shifter components.

Early array based radar systems used to perform pattern synthesis by means of ana-log components, such as phase shifters1_{(PS) or variable time delay lines (TDL) [2, 3],}

electronically controlled by a computer unit. As shown in Fig.1.1, the signals are then summed at the receiver level, which yields a single beam, and digitally converted be-fore being available to the radar signal processor. The choice about using either PS or TDL, aside from the inherent cost of the systems based on the latter ones, is driven by the transmitted pulse characteristics with respect to the array fill time [4]. In order to really benefit from all the array elements, the signal has indeed to be present on each of them before summation. If TDL are considered, this condition can be satisfied by a proper adjustment of each acquisition channel delay. In the case of PS, the signal duration has to be greater than the time needed by the electromagnetic wave to cover the array extension. By considering a linear array geometry, about which more details are provided in Ch.2, it can be written that:

τ T = L sin θ

c , (1.1)

being L the array length, θ the angle of arrival of the signal and c the speed of light. The quantities τ and T represent the pulse duration and the array fill time respectively. Since the pulse duration is inversely proportional to its bandwidth B, by means of a proportionality term kB, (1.1) can also be written as:

B c

kBL sin θ

. (1.2)

1_{The wording phased arrays was indeed introduced to refer to these type of analog systems.} Nowadays it is kept and it is more generally referred to any class of antenna arrays which perform signal processing at an element level.

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3

Figure 1.2: Basic digital beamforming scheme.

If we divide both sides of (1.2) by the carrier frequency fc, we obtain:

B fc

λ

kBL sin θ

, (1.3)

where λ is the wavelength. Expression (1.3) highlights the dependency between the signal fractional bandwidth _fB

c and the array system characteristics. If (1.3) is

satis-fied, the system is considered operating under narrow band assumption and PS can be used in place of TDL, which are then exploited in the case of wideband arrays. Thanks to the evolution of the radar electronics, state of the art systems can apply digital conversion at an element level (see Fig.1.2). In this case, each receiving channel has an independent front-end and an Analog to Digital Converter (ADC). Maximum freedom is then provided to the signal processing section for the application of the digital processing schemes, that are conventionally referred to as Digital Beam Form-ing (DBF) algorithms. In the digital domain, the antenna array can electronically be steered to any direction. Multiple and closely spaced antenna beams can be syn-thesised and independently pointed and shaped. A more accurate control of both the sidelobes behavior and the directional nulling is obtained thanks to calibration techniques that can directly be implemented in the digital domain [2]. With respect to surveillance operations, the DBF processing allows following the trajectory of the detected targets by means of narrow beams, while the sensed area being illuminated with a wide beam. Since the two operations are carried on at the same time, on the one hand, the performance of the radar tracker is improved, on the other hand, there is no reduction of the update rate of the sensed area. The electronic steering

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capability also represents an essential requirement in case of wideband and/or near field applications2_{[5, 6].}

Latest developments of sensing techniques have led to the realisation of novel radar concepts for security, safety and surveillance applications based on phased array tech-nology [7–9]. A growing interest in that sense is represented by the definition of proper array processing approaches for passive and Multiple-Input Multiple-Output (MIMO) radars.

1.1 Passive Coherent Locator (PCL) systems

Active monostatic radars work by transmitting a known signal and receiving the reflection of illuminated objects. As a consequence, while they detect a target, they can also be detected by hostile sensing systems because of their own transmission. To improve the covertness of the operation, bistatic systems were deployed already in the early days of radar development, with the transmitter located far from the receiver. While such a configuration allowed protecting the receiver, the transmitter could still be detected and damaged by the enemy. Thanks to the use of transmitters of opportunity, such as radio, mobile phone and TV broadcasters, a Passive Coherent Locator (PCL) does not produce any emissions and is inherently covert.

The target location is retrieved in terms of a bistatic range, which is the sum of the distance from the transmitter to the illuminated object and from this object to the receiver. Given the bistatic range, the object is located on an ellipse having as focal points the transmitter and the receiver. To resolve the object position, the direction of arrival of the reflected signal is needed. For this purpose, passive radars typically contain an antenna array and the target bearing angle is retrieved by means of array signal processing [10]. A crucial issue which influences the system performance relates to the capability of separating the relatively small signal reflected by the object and the direct signal produced by the emitter of opportunity. The large difference in strength between these two signals makes the design of passive radar system rather cumbersome [11]. Furthermore, the accuracy also depends on the emitted waveform and the related ambiguity function characteristics [12].

The covertness characteristic and the bistatic observation geometry, the latter one particularly suitable against stealth targets [13], have stimulated the research about PCL for military purposes. More recently air and vessel traffic control applications have also been considered [14–16].

Experimental PCL programs have been developed in several research institutes and industries in the world. Among them, the Lockheed Martin Silent Sentry in USA, the french HA-100 developed by Thales, the CORA PCL system designed at the

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1.2 The Multiple-Input Multiple-Output radar concept 5

Fraunhofer institute in Germany and the italian AULOS realised by SelexES. TNO -Defence, Safety and Security has been actively working on passive radar systems since 2002 [17]. Different experimental systems have been designed and manufactured for research purposes. The latest demonstrator is based on a circular array configuration that, with respect to linear array systems, offers the benefit of an omnidirectional coverage [18]. As a consequence, the direct signal is always present since there is not a physical separation between the channels which are aimed at acquiring the surveillance and the reference signals. The two channels are indeed synthesised by means of DBF techniques, that are also used to perform a preliminary suppression of the direct signal. With respect to this system, the following research objectives have been addressed:

• The identification of a proper calibration technique for the PCL system, ac-counting for both the analog and the digital sections of the receiver. For the application of any DBF scheme, it is indeed essential to have a complete under-standing of the antenna configuration and the RF chain of the radar.

• The derivation of a novel DBF method for the synthesis of the reference and the surveillance beams. Since the target direction of arrival estimation depends on the angular resolution provided by the array, this parameter has represented a driving point for the selection of the preferred approach. A comparison with existing technique, aimed at showing the benefits of the proposed method, has been carried out. The effective direct signal nulling capability that is achievable by means of DBF has also been evaluated.

• The assessment of the PCL system by means of real data acquisition and per-formance validation. To that aim, the entire signal processing chain has been implemented and the detection performance compared with ground truth data sets.

1.2 The Multiple-Input Multiple-Output radar concept

Unlike phased array radars, where the radiating elements transmit a scaled version of the same waveform, the MIMO paradigm is based on the possibility of employing multiple emitters in order to transmit probing signals that are orthogonal to each other [19]. At the receiver side, these uncorrelated waveforms can be separated to enable processing of each independent transmitter/receiver pair.

The orthogonality among the probing signals can be achieved in different domains [20]. The most obvious way is by allocating each emitter to a separate time slot, in the so called Time Division Multiplexing (TDM); drawbacks of this method are the scarce capability of dealing with fast moving targets and the increased requirement in data

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handling (the transmitted waveforms must separately be digitised). Another approach is based on the Frequency Division Multiplexing (FDM), in which case each waveform belongs to a different sub-band. Here the main issue is represented by the difficulty to ensure the desired coherency between pulses which are located at different frequencies. The last category is represented by the Code Division Multiplexing (CDM), that is an inheritance of the wireless communications where the MIMO paradigm was firstly applied. In this case digital codes with low cross-correlation profiles are exploited to modulate the basic waveform.

MIMO radars can also be classified on the basis of the transmitters and receivers spatial distribution. The case of widely separated radiators is normally referred to as statistical MIMO [21]. Such systems exploit the intrinsic angular diversity of the system layout in order to obtain uncorrelated reflections of the target. The process-ing of the multiple responses helps in mitigatprocess-ing the rapid signal fluctuations of the target Radar Cross Section (RCS). An improved target detection and estimation per-formance is then achieved [22, 23]. The other class of MIMO radars, namely coherent MIMO radars, is denoted by the presence of antenna elements placed in proximity to each other. By assuming each transmitter/receiver couple excited by the same target scattering response, i.e. the target response keeps its coherency among the multiple transmitted waveforms [24–26], the different contributions can be coherently combined leading to the synthesis of an extended number of virtual array channels. Array signal processing techniques based on the exploitation of these additional chan-nels ensure a better angular resolution of the radar system and an increased number of detectable targets [27].

The RADOCA radar demonstrator, where RADOCA stands for RAdar DOme CAm-era, is the result of one of the research projects led at TNO in the framework of coherent MIMO systems. The idea consists of combining a camera with a radar in order to perform the detection and classification of slow moving targets in private environments. To this aim, the radar has to provide the information (azimuth and range) about the target to the camera, that can then be steered towards the object and finalize its identification. Design requirements in terms of: system size, angular resolution and spatial adaptivity for clutter mitigation have driven the choice to the realisation of a TDM based MIMO radar. The research activity within this framework has focused on:

• The theoretical characterisation of the coherent MIMO radar concept by the as-sessment of advantages and disadvantages with respect to conventional phased array based radars. Specifically, a proper dimensioning of the MIMO system, in both ideal and non ideal conditions, has been studied and the expected perfor-mance for the angular resolution and the array pattern sidelobes behavior have been retrieved.

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1.3 Outline of the Thesis 7

• The estimation of a calibration procedure aimed at compensating for the array element mismatching. As in the case of phased arrays, the application of DBF techniques and the correct behavior of the radar depend on a proper calibration of the array antenna section. An approach that exploits the MIMO configuration has been studied to this purpose.

• The analysis of the limiting factors which are introduced by the selected TDM scheme. Their impact on the target identification have been analysed and a transmitter selection procedure aimed at overcoming this limitation has been proposed.

• The evaluation of high resolution array processing techniques for radar perfor-mance enhancement. Thanks to the exploitation of the extended number of degrees of freedom provided by the coherent MIMO array, a novel technique has been implemented and the better range/angle discrimination capability has been assessed.

1.3 Outline of the Thesis

The remainder of the thesis is organised as follows:

Chapter 2 introduces the principles of general antenna and array antenna theory. The main concepts to assess and to describe the array pattern behavior are presented. Causes of pattern degradation, i.e. mutual coupling and element il-lumination errors, are explained and referred throughout the text to characterise the radar systems under investigation.

Chapter 3 focuses on the algorithm for the mutual coupling compensation of the circular array which is used as a PCL sensor. A set of internal and external measurements are used for the compensation procedure and the related signal processing operations are illustrated. An array pattern shaping technique is also illustrated. The performance of the proposed method is then compared with the phase modes technique and the specific benefits are highlighted.

Chapter 4 presents the PCL radar that has been the object of the digital beamform-ing research on circular arrays. The antenna section of the system is analysed and the design choices of the receiver are depicted. The techniques illustrated in Ch.3 are then applied and assessed by means of experimental validations. Also, the entire signal processing chain, leading to target detection and plot ex-traction, is presented and the radar performance is compared with Automatic Dependent Surveillance Broadcast (ADSB) ground truth data.

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Chapter 5 discusses the topic of digital beamforming in connection with the coher-ent MIMO theory. The procedure leading to the design of this kind of radar is presented and the concept of waveform diversity is introduced. Then, the char-acteristics of the MIMO array pattern and the degradation due to the presence of element illumination errors are addressed. The higher sensitivity to the illu-mination errors of the MIMO arrays, with respect to conventional linear arrays, is described and proven by the retrieved analytical formalism.

Chapter 6 describes the RADOCA radar demonstrator that has been realised and the results of the signal processing techniques which have been developed: the calibration procedure, the transmission approach aimed at extending the unam-biguous Doppler interval for the TDM based MIMO radars and the novel high resolution technique. Both simulated and experimental results are presented and discussed.

Chapter 7 summarises the main achievements of the research activity described in this thesis and provides an overview of the challenges which can still be the objective of further analyses, both from hardware and signal processing perspectives.

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Chapter 2 Antenna theory and array

pattern synthesis

The chapter is concerned with the description of the main antenna and array antenna pa-rameters that will be used throughout the text. In the first section, the basic definitions of the antenna radiating properties are considered. Then, specific attention is given to the in-troduction of the array pattern concepts with respect to the linear and the circular shape configurations. With the aim of characterising the effect of the non ideal behavior of the array elements, the consequences due to the presence of mutual element interactions and aperture illumination errors are considered in the last section of the chapter.

2.1 Main antenna parameters

The classical definition of antenna is provided by the “IEEE Standard Definitions of Terms for Antennas (IEEE Std 145-1983)” [28] where it is defined as “the part of a transmitting or receiving system which is designed to radiate or to receive electromag-netic waves”. It then results to be the transitional structure between the free-space and the guided propagations of an electromagnetic wave. From the circuital point of view, by exploiting the Thevenin representation of a guided propagation channel, the antenna can be described by means of the impedance [29]:

ZA= (RL+ Rr) + jXA, (2.1)

being RL the resistance associated to the dielectric and conduction losses of the

an-tenna structure, Rrthe radiation resistance and XAis the reactance which represents

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under ideal conditions, the conjugate matching between the antenna characteristic impedance and the internal impedance of the source. In this way, the maximum power is delivered to the antenna.

Besides its function as a receiving or transmitting energy device, most of the appli-cations require the optimization of the radiated energy into some specific directions, while it has to be minimized into others. Here the main parameters which are used to describe the directional behavior of an antenna are introduced, since they will be referred to later in the thesis. A more detailed and extensive description can be found in [29–32].

2.1.1 Directivity

The directivity function D(θ, φ) of an antenna is defined as the ratio between the radiation intensity U (θ, φ)1 _{in a given direction and the radiation intensity averaged}

all over the directions. Being the average radiation intensity equal to the total power radiated by the antenna divided by 4π, it follows that

D(θ, φ) = U (θ, φ) U0 = 4πU (θ, φ) Prad , (2.2) where U0 = 4π1 R

ΩU (θ, φ)dΩ and Ω is the solid angle. From (2.2) it is clear that, in

the case of an isotropic radiator, the directivity is unity as the quantities U (θ, φ) and U0 are equal to each other.

Under the far field assumption2_{, the directivity can also be expressed in terms of the}

electric field component as:

D(θ, φ) = 4πE(θ, φ) 2 R ΩE(θ, φ)2dΩ . (2.3)

2.1.2 Efficiency

A number of efficiencies can be related to an antenna; the overall efficiency η is normally taken as the combination of all of them and it is written as:

η = ηrηcηd, (2.4)

where ηr is the reflection efficiency, which depends on the mismatch between the

transmission line and the antenna, ηc and ηd are the conduction and the dielectric

1_{The radiation intensity represents the power radiated by an antenna per unit solid angle [30].} 2_{The far field region is assumed at a distance greater than} 2D2

λ being D the diameter of the sphere which includes the antenna and λ the wavelength. This approximation keeps its validity at the frequency bands of VHF and above. At lower frequencies, the starting distance of the far field region is dictated by the most stringent value between 10D and 10λ.

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2.2 Antenna array pattern synthesis 11

efficiencies respectively and they are related to the dissipation losses.

The efficiency represents a quality factor of the antenna performance as it indicates how much of the input power Pin is effectively radiated by the antenna:

Prad= ηPin. (2.5)

2.1.3 Gain

A parameter which takes into account both the directional and the efficiency prop-erties of an antenna is the gain, which is defined as “the ratio between the radiation intensity in a given direction and the radiation intensity that would be obtained if the power accepted by the antenna were radiated isotropically”. Being the radiation intensity corresponding to an isotropic radiator equal to the input accepted power divided by 4π and by considering (2.5) we have:

G(θ, φ) = 4πU (θ, φ) Pin

= η4πU (θ, φ) Prad

. (2.6)

The relation between the antenna gain and the antenna directivity can then be written as:

G(θ, φ) = ηD(θ, φ). (2.7)

The gain is then equal to the directivity in the case of an ideal radiator which presents no losses, i.e. efficiency equal to unit. It must also be said that the IEEE standard definition does not usually include the reflection efficiency in (2.7), limiting therefore the efficiency to the dielectric and the conduction contributions.

2.2 Antenna array pattern synthesis

Multiple independent radiating elements can be arranged into a certain geometrical configuration in order to improve the performance of the transceiver section. These antenna configurations are usually referred to as “arrays”. The main advantage pro-vided by the antenna arrays is represented by their electronic scanning capability. Furthermore, an opportune tapering of the excitation coefficients of the different el-ements can be used to control the shape and the sidelobes of the array pattern as it will be shown in Sec.2.2.1 and Sec.2.2.2. The array pattern control also conceives the possibility of simultaneously steering the main beam in a certain direction while a null is placed towards the direction of an interfering/jamming signal.

The behavior of the array pattern mainly depends on five parameters [29]: • geometry, i.e. the spatial configuration of the array

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Figure 2.1: Generic antenna array geometry.

• spacing between the multiple elements • amplitude excitation of each element • phase excitation of each element • element pattern of the radiators

The resulting electric field excited by a radiating element in a given point of the space depends on the distance from the element itself and the related angular coordinates. If the point is taken at a very far distance from the element, then the electric field can be represented as the product between an angular function, fn(θ, φ), which is

referred to as the element pattern and a radiation term which has spherical wave dependance with the range [4]. By considering the geometry illustrated in Fig.2.1, which represents a generic distribution and composition of array elements, the electric far field produced the by the n-th element can be written as:

En(θ, φ, r) = fn(θ, φ)

e−jkRn

Rn

, (2.8)

being k the free space wave number and: Rn=

p

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the distance between the n-th array element and the point where the electric field is measured. If this distance is considered in the far field of the array3 _{the related}

electric field wavefront can be assumed to be planar and (2.9) can be approximated by:

Rn= R − ˆr · ~rn, (2.10)

where ˆr is the versor to the point where the electric field is measured and ~rnis position

vector of the n-th array element. The origin of the reference system can be arbitrarily chosen. According to (2.10), the expression in (2.8) can be rewritten as:

En(θ, φ, r) = fn(θ, φ)

e−jkR R e

jk(ˆr·~rn)_. _(2.11)

The total electric field can then be retrieved by superposition of all the element contributions: E(θ, φ, r) = e −jkR R N X n=1 fn(θ, φ)ejk(ˆr·~rn). (2.12)

Since the angular properties of the electric field in (2.12) are measured on a sphere of constant radius, the term e−jkR_R can be neglected and (2.12) can be written as a function of the angular variables only. Thus, if all the elements share the same pattern characteristic and different complex excitations an are considered for the

multiple elements of the array, (2.12) becomes:

P (θ, φ) = f (θ, φ)

N

X

n=1

anejk(ˆr·~rn) (2.13)

which represents the array pattern. The aim of the an tapering coefficients is to

perform both the array pattern shaping and steering. More details about the weights selection are provided in Sec.2.2.1 and Sec.2.2.2 with respect to linear and circular array configurations. These geometries are of particular interest since they are the chosen structures for the design of the radar systems that are illustrated in Ch.3 and Ch.6.

According to (2.13), the array pattern can be seen as the product between two main components: the element pattern and the array factor :

F (θ, φ) =

N

X

n=1

anejk(ˆr·~rn) (2.14)

which describes the geometrical characteristics of the array.

3_{The array far field region is considered to start at a distance R =} 2L2

λ , being L the largest dimension of the array. However, it is pointed out in [4, 33, 34] that, in order to measure very low sidelobes and array patterns with deep nulled regions, the more stringent limit of R =10L2

λ should be taken as reference.

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2.2.1 Linear array pattern synthesis

The linear array structure, or uniform linear array (ULA) when the an coefficients

have the same amplitudes, foresees a deployment of the antenna elements on a straight line, as it is shown in Fig.2.2 for an array of N elements. According to this geometry, the steering capability of the array is limited to the xOz plane (φ = 0), whereas the angular performance on the xOy plane is dictated by the behavior of the element pattern only.

The analysis of the array factor for this configuration can be done by taking for simplicity a phase reference at the first element, then (2.14) takes the form:

F (θ) =

N −1

X

n=0

ane−jknd sin (θ), (2.15)

where d is the inter-element spacing between the radiators. The series in (2.15) has a known expression4 and it can then be rewritten as (the tapering coefficients are assumed equal to one):

F (θ) = 1 − e −j2π λN d sin(θ) 1 − e−j2π λd sin(θ) = e −jπ λN d sin(θ) e−jπ λd sin(θ) ejπλN d sin(θ)− e−j π λN d sin(θ) ejπ λd sin(θ)− e−j π λd sin(θ) = ejφrefsin π λN d sin(θ) sinπ λd sin(θ) (2.16)

Figure 2.2: Linear antenna array geometry scanning on the xOz plane.

4PA−1 a=0 e

−jax₌1−e−jAx 1−e−jx .

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and it is also valid:

F (θ) = ejφrefsin π λN d sin(θ) sinπ_λd sin(θ) = e jφref_Nsin π λN d sin(θ) N sinπ_λd sin(θ) ≈ ejφref_{N sinc} hπ λN d sin(θ) i , (2.17)

where the contribution of the array gain N , which depends on the total elements number, has been highlighted.

When the complex excitations are chosen to synthesise the main beam towards a specific direction θ0, which is obtained by selecting an= ejknd sin(θ0), it can easily be

retrieved that (2.16) becomes:

F (θ) = ejφrefsin π λN d(sin(θ) − sin(θ0)) sinπ λd(sin(θ) − sin(θ0)) . (2.18) Grating lobes

By analysing (2.18) it is clear that the element contributions can coherently add up together at different angles. Those angles correspond to the zeros of the denominator:

π

λd| sin(θ) − sin(θ0)| = pπ with p = 1, 2, ... (2.19) and, as a result, multiple beams of the same amplitude, usually referred to as grating lobes, can arise in the field of view of the array. This occurrence can be avoided by properly choosing the inter element spacing d. For a given maximum pointing angle θ0, if only a single beam, i.e. the main beam, has to be synthesised, the following

condition must hold:

d λ< N − 1 N 1 1 + | sin(θ0)| , (2.20)

which is a conservative constraint as it imposes that the entire grating lobe must be outside of the scanning region of interest.

Beamwidth and sidelobes

The pattern beamwidth, which corresponds to the physical angular resolution of the array, can be referred to either the Half Power Beamwidth (HPB) or to the First Null Beamwidth (FNB). The former is measured at the -3dB intersection points from the array pattern main peak, which for large arrays5 _{and by considering the boresight}

5_For _a _generic _array _length _the _expression _which _provides _the _3dB _beamwidth _is: sin(πN d

λ sin(θ3dB)) N sin(πd

λ sin(θ3dB))

=√0.5. However, it is shown in [4] that for N ≥ 3 the beamwidth variation with respect to (2.21) is less than 5% and it becomes lower than 1% for N ≥ 7.

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pointing direction are given by: θ3dB= sin−1 ±0.443 λ N d , (2.21)

the latter, by considering (2.18), is given by the solution of: π

λd(sin(θ) − sin(θ0)) = π

N (2.22)

which after few steps6_{results to be:}

θF N B= sin−1

_λ

N d cos(θ0)

. (2.23)

Equation (2.23) also highlights the resolution degradation which is introduced by the pattern steering. By combining (2.21) and (2.23), the relation between the HPB and the FNB is retrieved:

θHP B ' 0.886θF N B. (2.24)

A useful descriptor of the array pattern behavior is the Side Lobe Level (SLL), which represents the ratio between the highest side lobe and the main beam amplitudes. In the case of large arrays, the SLL is independent of the main beam angle [35] and, if a uniform excitation is applied throughout the elements, a -13.2dB level is obtained. The SLL can be decreased by applying non uniform tapering masks at the cost of both a reduced illumination efficiency:

ηi= 1 N (PN n=1|an|)2 PN n=1|an|2 (2.25) and a degraded angular resolution. Any novel technique that is implemented in order to improve the SLL of an antenna array should also be analysed by taking into account these effects.

Fig.2.3 provides an example of all the concepts that have just been introduced. Three normalised pattern behaviors, belonging to a 16 elements linear array, are illustrated. The first plot shows the uniform illumination case with an inter-element spacing d = λ/2. The FNB and the HPB are depicted. In the second curve, while keeping the uniform illumination, the array spacing has been increased to the value of d = λ leading to the synthesis of a grating lobe at θ = 90o_{. In the last plot, a Chebishev}

tapering mask [36] providing an SLL=-30dB has been applied instead of a uniform one. The broadening of the main beam can be noticed and a resulting illumination efficiency ηi= 0.86 is obtained.

6 πd

λ [sin[(θ − θ0) + θ0]] ≈ πd

λ [sin(θ − θ0) cos(θ0) + sin(θ0) − sin(θ0)] =πd_λ sin(θF N B) cos(θ0).

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Figure 2.3: Normalised array pattern behaviors along the azimuthal plane (θ = 90o).

2.2.2 Circular array pattern synthesis

The main advantage of using a circular array configuration is represented by its sym-metry on the plane where the array is deployed. As a result, the pattern characteristics do not present an high variation while the beam is electronically steered. For this reason, the circular arrays are particularly suited for applications that require a 360o coverage as it is for direction finding systems, auxiliary antennas, communication bridges and so on.

In Fig.2.4 the typical geometry of an N elements circular array is shown. By follow-ing similar steps to the ones which have led to the representation of the array factor for the linear array case, the Rn distance between the n-th array element and the

observation point in the far field region is equal to: Rn=

p

R2_{+ a}2_{− 2aR cos ξ}

n, (2.26)

being a the array radius and R the distance between the array center and the point where the electric field is measured. Considering that R a, Rncan be approximated

in the following way:

Rn ' R − a cos ξn = R − a(ˆar· ˆan)

= R − a(ˆaxcos φn+ ˆaysin φn)

·(âxsin θ cos φ + âysin θ sin φ + âzcos θ)

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x y z 1 ₂ n

R

n R f a an ar fn xn q

Figure 2.4: Circular antenna array geometry.

where φn= 2π_N(n − 1).

The complex excitation coefficients that are used to perform both the shaping and the steering of the array pattern can explicitly be written as:

wn= Inejαn. (2.28)

By considering (2.15), (2.27) and (2.28), the array factor in the circular array case is then given by:

F (θ, φ) =

N

X

n=1

Inej[ka sin θ cos(φ−φn)+αn]. (2.29)

The synthesis of the array pattern towards a specific direction (θ0, φ0) can be obtained

by selecting αn= −ka sin θ0cos(φ0− φn) and it consequently follows that:

F (θ, φ) =

N

X

n=1

Inejka[sin θ cos(φ−φn)−sin θ0cos(φ0−φn)]. (2.30)

The pattern of the array factor for an 8-elements Uniform Circular Array (UCA), having therefore ∀In = 1, is presented in Fig.2.5, whereas the cuts along the azimuthal

and the elevation planes are shown in Fig.2.6 The first sidelobe in the UCA case shows a relative amplitude of −7.9dB, which is the typical value for this class of arrays. A traditional way to perform the analysis and the pattern shaping of UCA systems is

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Figure 2.5: 3D view of the array factor for an 8-elements UCA (linear scale).

Figure 2.6: Array factor for an 8-elements UCA along the azimuthal (θ = 90o_{, left) and the elevation} (φ = 0o_{, right) planes.}

based on the phase modes theory, which will be presented in Sec.3.3.1. By exploiting that representation, in [37,38] an extensive analysis of the parameters selection, which allows avoiding distortions of the synthesised pattern, is conducted.

Specifically in [38], the condition to prevent the grating lobes is proven to be:

d < λ

2 (2.31)

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2.3 Pattern synthesis in non ideal arrays

The array pattern analyses, which have been presented in Sec.2.2.1 and Sec.2.2.2, are based on the ideal behavior of the array manifolds. However, real arrays are affected by different sources of errors that introduce degradations in terms of sidelobes level, pointing accuracy and angular resolution of the synthesised pattern. Although it is possible to compensate for some of these errors, as it is described in Sec.2.3.1 and with more details in Sec.3.2, proper design choices are required in order to overcome the physical limitations produced by the random error distributions of the system, which cannot be corrected. This case is analysed in Sec.2.3.2 and it is extended in Ch.5 for a particular type of arrays.

2.3.1 Mutual coupling

The expression retrieved in (2.13) is based on the assumption that the element pattern is the same for all the radiators. However, the gain of an isolated element may substantially vary when the same element is placed inside an array. This gain variation also depends on the specific location of the radiator: the behavior at the edges of the structure is quite different from the one in the center. For this reason, a reliable evaluation of the array pattern behavior cannot be exempted from the assessment of the mutual electromagnetic influences, which are usually referred to as Mutual Coupling (MC) [39, 40], among the array radiating elements.

The element pattern affected by MC can be written as the product between the expected ideal pattern fi(θ, φ) and a spatial factor that accounts for the coupled

element contributions [4]: fn(θ, φ) = fi(θ, φ) " 1 +X m Snme−jk(~rm−~rn)ˆr # , (2.32)

being Snm the scattering coefficient [41]. The pattern in (2.32) is measured in a

controlled environment and by transmitting with the n-th element while all the other elements are closed on matched loads. The total coupling effect is quantified by collecting all the scattering contributions in the so called MC matrix [42, 43], which is then used to retrieve the ideal behavior of the array pattern [44–47]. Due to its dependance on several parameters, it is preferable to perform the direct measurement of the MC matrix instead of predicting its coefficient values. In Ch.3 a novel technique aimed at estimating the MC matrix is presented with respect to the case of a circular array in an interfering environment.

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2.3 Pattern synthesis in non ideal arrays 21

2.3.2 Illumination errors

An array illumination distribution, which is aimed at providing a very low sidelobes array pattern, may result in a poor SLR because of the presence of amplitude and phase errors introduced by the array imperfections [2, 4]. This type of errors, which are caused by the tolerance and quantization limits of the array devices (feeding network, phase shifters, element orientations and positions, ...), can be treated as a random and uncorrelated process7_{. The sidelobes level degradation is produced by}

the random sidelobes that are generated and that add to the expected ones.

If the errors for the N elements array are assumed to be independent but identically distributed, and by referring to the linear array case, the AF affected by illumination errors can be written from (2.15) as:

Fe(θ) = N −1

X

n=0

an(1 + δn)e−j(knd sin θ+δφn), (2.33)

where δn and δφn are the fractional amplitude and phase errors associated to the

n-th element respectively. The AF is now a random variable and it depends on the statistical distributions of δn and δφn. The mean value of the AF is:

hFe(θ)i = N −1 X n=0 ane−jknd sin θe−jδφn + N −1 X n=0 hδni e−jknd sin θe−jδφn , (2.34)

where the second addend in the sum is equal to zero if the presence of systematic errors is avoided, i.e. the random processes are not biased. The first contribution represents the product between the ideal AF and the characteristic function8_{evaluated in ω = 1.}

For a uniform phase error distribution: p(δφn) =

1

∆φrect∆φ(δφ), (2.36)

(2.33) can then be written as:

hFe(θ)i = F (θ)

sin(∆φ)

∆φ . (2.37)

With respect to the field intensity, which is proportional to the array factor as shown in (2.13), it is clear from (2.37) that the effect of the illumination errors only consists

7_{For an extended analysis, the correlated errors case could also be considered and it would result} in higher sidelobes at fixed locations [2].

8_{The characteristic function is defined as:}

C(ω) =e−jωx₌Z _e−jωx_px(x)dx. _(2.35)

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of a little attenuation of the expected ideal pattern.

In the case of the power pattern, which is related to the radiation intensity through (2.2)-(2.3), its representation can be retrieved by taking the average value of the product between (2.33) and its complex conjugate value:

D |Fe(θ)|

2E

= hFe(θ) · Fe∗(θ)i . (2.38)

The solution of (2.38) has been firstly retrieved in [48], then further investigated in [49, 50]. Here the final expression is provided:

D |Fe(θ)| 2E = |F (θ)|2+ δ2 n + δφ 2 n ηi. (2.39)

An extension to the case of virtually synthesised arrays is retrieved in Ch.5, where also more details about the analytical solution are given. The average power pattern that is shown in (2.39) is the sum of two contributions: the ideal power pattern and a term depending on both the amplitude and the phase error distributions. It can be observed that it is independent of the angular coordinates whereas it is directly proportional to the illumination efficiency ηi. According to the mentioned

character-istics, the second term in (2.39) introduces a uniform rise of the sidelobes level all over the angular domain of the pattern; the effect is however of minor relevance in the main beam and in the near sidelobe regions [49].

It is also of interest to observe that, by referring to (2.25) which shows a 1/N depen-dency with the array elements number, the larger the number of elements, the lower will be the degradation of the sidelobes produced by the illumination errors.

2.4 Summary

The aim of this chapter was to provide the basic theoretical notions that allow de-scribing the behavior of an antenna in both the isolated and the array configuration cases. Attention has been given to the characterisation of the array pattern synthesis for the linear and the circular array structures. At first, the ideal pattern behaviors have been retrieved and the expected performance in terms of beamwidth, charac-teristic sidelobes level and shape have been discussed. Secondly, the main causes of pattern degradation at both the antenna level (MC) and the system front end (illu-mination errors) have been presented. These analyses pave the way for the system investigations in Ch.3, Ch.5 and Ch.6 where real radar demonstrators based on the above mentioned configurations are considered.

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Chapter 3 Digital beamforming for PCL

The signal processing steps required by a radar system are multiple and they have to deal with the receiver chain at different levels. A first distinction can be done on the basis of the analog and the digital sections that are involved. The first one, which is mainly based on the front-end system architecture, will be analysed in the next chapter. Here the attention is focused on the description of the digital processing blocks that represent the back-end part of the system. A further distinction is required. The digital section can indeed be considered as composed by two different macro blocks: the one belonging to the DBF processing, which is tailored on the specific array characteristics, and the one related to the conventional signal processing steps of the radar system. In this chapter, the entire attention is given to the analysis of the implemented DBF procedures for the PCL system, which represents the core of the research activity. However, in order to fully characterise the system performance, also the remaining signal processing steps have been implemented and they will be shown in Ch.41.

3.1 Digital processing scheme

A sketch of the digital processing steps that are involved in the PCL system opera-tions is presented Fig.3.1. The sub-division into the two macro-blocks referred to as the DBF and the specific radar signal processing is also highlighted. With respect to the DBF part, the first operation is aimed at compensating for the MC interactions among the antennas of the circular array. The non ideal behavior of the receiver elements is corrected by means of the estimation of the MC matrix. This operation is considered as a part of the DBF section since it allows the application of the beam-forming algorithms, which otherwise would not provide the expected performance.

1_{This chapter is based on articles [J1], [C8], [C9] and [R1] (a list of the author’s publications} is included at the end of this dissertation, p. 141).

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Figure 3.1: Digital signal processing scheme of the PCL radar.

Specifically, a method for the sidelobes shaping is proposed and the related results are compared with a well established technique, which is especially tailored for cir-cular array patterns. The pattern shaping technique, in conjunction with the digital steering, is used to retrieve the reference and the surveillance channels of the system. While the surveillance beam is synthesised, a digital null is placed in the direction of the reference transmitter. The spatial nulling is part of the Direct Path Interference (DPI) suppression, which is aimed at reducing the jamming effect of the direct signal in the surveillance channel, and it represents the last step of the processing referred to as the DBF.

The overall signal processing chain is then completed by: a digital filtering operation, based on the Least Mean Squared (LMS) principle, which integrates the spatial DPI nulling; the Matched Filter (MF), based on the cross correlation between the surveil-lance and the reference channels; the Constant False Alarm Rate (CFAR) thresholding that provides the output target detections.

3.2 Mutual coupling compensation

Most of the MC compensation techniques require either a detailed knowledge of the electrical characteristics of the antenna [51], or the possibility to measure them in a controlled environment, e.g. an anechoic chamber, [52, 53]. The former case, as it is highlighted in Sec.4.2.1, cannot be considered due to the lack of information about the commercial components of the reference system; the latter one, due to the size of the entire structure, is not applicable either. The compensation technique that is proposed in this section, in order to overcome the mentioned issues, is based on measurements aimed at obtaining a preliminary estimation of the MC matrix. Then, a further refinement is obtained by means of an optimisation approach.

3.2.1 Analytical description

In real arrays, the element pattern is affected by the neighboring elements and, if we refer to a circular array, it is an angular function of the element position. By recalling

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3.2 Mutual coupling compensation 25

the characterisation presented in Ch.2, the far-field array pattern for a N-element circular array can be written as:

P (φ) =

N

X

n=1

anfn(φ − φn)ejkr cos(φ−φn). (3.1)

The dependency with the elevation angle θ is omitted for simplicity. Similarly to [42], the total received voltage at the m-th element can be written as a weighted sum of the contributions of all the array elements:

vm(φ) = cmmEmfi(φ − φm) + N X n,n6=m cmnEnfi(φ − φn) = N X n=1 cmnEnfi(φ − φn), (3.2)

being Emthe electric field which excites the m-th element port and fithe ideal element

pattern. The mutual coupling coefficient cmntherefore represents the proportionality

term that relates the induced voltage on channel m to the total voltage on channel n. The desired voltage on channel m is then given by:

vmd(φ) = Emfi(φ − φm). (3.3)

If we compare (3.2) and (3.3) and by taking into account that Em= Em0e

jkr cos(φ−φm)

we obtain the same expression as [42] for the linear case which relates the desired voltage signal and the real (affected by mutual coupling) one. By means of matrix notation it can be written as:

     v1 v2 .. . vN      =      c11 c12 . . . c1N c21 c22 . . . c2N .. . ... ... ... cN 1 cN 2 . . . cN N      ·      vd 1 vd 2 .. . vd N      , (3.4)

or in a more compact way as:

vd= C−1v. (3.5)

The MC matrix C can be written as a function of the antenna scattering parameters S [4, 41]:

C = S + I, (3.6)

where I is the identity matrix.

The evaluation of the coupling coefficients in (3.5) can be performed in an analytical or numerical way, depending on the antenna type and the array configuration [29].

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If we refer to real systems, the effect of the feeding network and the analog/digital front-end must also be considered, since it introduces discrepancies with respect to the conventional representation. By considering the cables section as a reciprocal and isolated structure, its contribution on the MC matrix can be modeled by a diagonal matrix T of the transmission coefficients2_{. From (3.6) we have:}

C = T(S + I). (3.7)

Equation (3.7) fully describes the matrix term which must be applied in order to compensate for the non ideal behavior of the analog section of the system. Since the presented correction is applied in the digital domain, e.g. after digital conversion of the received signals, the effect of the phase and amplitude errors introduced by the digital receiver should also be taken into account. The final correction matrix can then be expressed as the following product:

C = P[T(S + I)], (3.8)

where P is the digital section compensation matrix.

3.2.2 Optimisation approach for the C matrix evaluation

The expression retrieved in (3.8) is valid when no other signal sources, which can generate local interferences to the radiation pattern, are present at the array location. However, the environment in which the TNO passive radar is located, the top of the tower at The Hague laboratory, is heavily affected by multipath due to the presence of several large metallic structures. Moreover, the FM signals coming from the local radio stations produce additive interferences and, as a consequence, an inaccurate measurement of the main components of the MC matrix. For this reason, the C matrix coefficients have been refined by means of an optimisation approach.

As proposed in [54], a reference transmitter has been moved around the PCL, at distances ensuring the far field condition, and the signals from the known positions have been acquired. For a far field monochromatic source, the received data vector can be written as:

x(t) = C[sa(t) + n(t)], (3.9)

where a(t) is the amplitude of the received signal and n(t) the noise realisation vector. The quantity s represents the steering vector of the array that, on the azimuthal plane and for an angle of arrival φm, takes the form:

s = [ejkr cos(φm)_{, e}jkr cos(φm−φ1)_{, ..., e}jkr cos(φm−φN −1)_]T_, _(3.10) 2_{The transmission coefficients represent the attenuation and the phase offset values to which the} signals propagating in the cables are affected.

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3.3 DBF for circular arrays 27

where (·)T _{is the transposed operator. As mentioned before, the φ}

mvalue is known.

By considering M transmission points, all of them characterised by the same C matrix, the following objective function can be written:

O(C, a1, a2, ..., aM) = M

X

m=1

kxm− Csmamk2 (3.11)

which represents a non linear system of equations whose unknowns are the signal am-plitudes [a1, a2, ..., aM] and the coupling matrix elements in C. The xmvector is the

collected data vector having the reference transmitter at the m-th location around the array. A Broyden-Fletcher-Goldfarb-Shanno (BFGS) Quasi Newton method [55, 56] has been applied to minimise the objective function in (3.11). In order to facili-tate the optimisation process, the values of C have been initialised according to the preliminary estimation provided by (3.8).

3.3 DBF for circular arrays

In passive radar applications the requirement on the SLL can be a very stringent constraint, when considering that the ratio between the reference signal and a target echo can be in the order of 90dB or more. Thus the patterns obtained in the previous section are not able to satisfy it. A further reduction can be achieved by means of a non uniform tapering on the array channels. Several DBF techniques for circularly shaped phased arrays are based on phase mode excitations theory [57–60]. Such theory aims at synthesising a virtual uniform linear array that can then be weighted with conventional tapering functions. However, the synthesis of the linear array results in a degradation of the pattern beamwidth and often in an illumination efficiency loss. If the second consequence can still be tolerated in PCL radars, thanks to the power strength of the signal which are exploited, a further deterioration of the angular resolution of the system should be avoided. To this aim a novel sidelobes shaping technique has been developed. The regions of interest of the circular array pattern are identified by an optimisation mask, as it is proposed in [54]. Then, the closed form analytical expression for the optimal beamforming vector is retrieved. The phase modes theory is briefly presented in Sec.3.3.1 whereas in Sec.3.3.2 the proposed technique is discussed and the comparisons of the two methods are illustrated.

3.3.1 Phase modes theory

A well known technique for the array pattern shaping of circular and cylindrical arrays is based on the phase modes theory. The array pattern of a circular array is a periodic function in the interval [0, 2π] and this characteristic allows representing it in terms

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of a complex Fourier series. Referring to (3.1) we can write: P (φ) = ∞ X p=−∞ = Pp(φ) = ∞ X p=−∞ Cpejpφ, (3.12) being: Cp = 1 2π Z π −π P (φ)e−jpφdφ. (3.13)

Each term of the series in (3.12) is normally referred to as phase mode of the radiation pattern [1] and it has a 2pπ phase variation as φ varies from 0 to 2π. From (3.12) it follows that, in order to synthesise a directional far field pattern Pp(φ), a p-th order

phase mode ejpφ_{must be produced in the far field. The C}

p coefficients can generally

be evaluated as a sum of the anexcitations, but that sum cannot be solved in a closed

form for all the an. However, a specific set of symmetrical excitation currents can be

identified [4] and computed as:

an = ∞ X p=−∞ apn = ∞ X p=−∞ Apejpφn. (3.14)

The terms of the sum in (3.14) are referred to as the phase mode excitation currents. By combining (3.1) and (3.14), the p-th phase mode pattern takes the form:

Pp(φ) = N

X

n=1

Apejpφnejkr cos(φ−φn). (3.15)

By introducing the Bessel function approximation for the array factor exponential term and neglecting a constant term of no interest, (3.15) can be rewritten as:

Pp(φ) ≈ ApjpJp(kr)ejpφ, (3.16)

where Jp(·) is the first kind Bessel function of order p. The behavior of the Bessel

functions, which is shown in Fig.3.2, also introduces a limitation on the maximum number of phase modes that can be excited on the array. Decreasing rapidly to zero when the order of the function exceeds its argument, the maximum phase mode order is estimated as the maximum value of the argument which is: P = bkrc.

The Ap coefficients can now be selected in order to synthesise the far field pattern of

the array. By choosing the coefficients as: Ap=

e−jpφ0

jp_J p(kr)

, (3.17)

the array pattern resembles the pattern of a linear array [57]. The only difference is that, whereas in the linear case the azimuth scanning angle φ is related to a sin(·)

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3.3 DBF for circular arrays 29 0 2 4 6 8 10 -0.5 0 0.5 1 J 0(x) J 1(x) J 2(x) J 3(x) J 4(x) J 5(x) X First kind Bessel functions

Figure 3.2: First kind Bessel functions of different orders.

function, providing therefore a possible steering sector of 180◦, in this case the scan-ning capability has no angular restrictions over the 360◦. This result is in accordance with the circular shape of the array.

Referring to the N excitations that must be applied to the circular array, they can be evaluated by combining (3.14) and (3.17) and by considering the upper bound of the effective number of phase modes that can be excited, it results:

an= P X p=−P ejp(φn−φ0) jp_J p(kr) . (3.18)

The tapering introduced by (3.18) produces a virtual transformation of the circularly shaped array into a linear array structure, resulting in a SLL of −13dB. A secondary effect resides in the possibility of over imposing a conventional tapering window in order to further reduce the SLL.

3.3.2 Proposed algorithm

The array factor of the UCA that has been introduced in Ch.2, can be written ac-cording to (3.10) in the compact form:

F (φ) = sHa(φ), (3.19)

where

a(φ) = [ejkr cos(φ), ejkr cos(φ−φ1)_{, ..., e}jkr cos(φ−φN −1)_]T _(3.20)

represents the array manifold and the superscript (·)H _{is the Hermitian operator. The}

(46)

Figure 3.3: Normalised UCA pattern and mask function fd(φ) of the desired pattern behavior.

regions: the sidelobe, the transition and the main beam regions. In accordance with the angular location of these sectors, it is possible to define a mask function fd(φ)

that describes a desired array factor behavior. An example is given in Fig.3.3 where the generic UCA pattern for an 8 elements array with r = 0.48λ is also plotted. The aim of a digital beamforming procedure is to identify the steering vector that, at the same time, minimises the contribution from the side lobes and focuses the energy into the main beam region without a considerable degradation of the angular resolution. If the mask function is taken as the objective of the pattern synthesis, the previous conditions can be expressed by the following minimisation:

O(s) = min

s {ks

H_{a(φ) − f}

d(φ)k2}. (3.21)

The effect of the minimisation in the sidelobes region can be emphasised by a diagonal matrix of weights Λ; thus (3.21) can be rewritten as:

O(s) = min

s {[s

H_{a(φ) − f}

d(φ)]Λ[sHa(φ) − fd(φ)]H} (3.22)

and by omitting the dependency with the angle φ: O(s) = min

s {s

H_aΛaH_{s − s}H_aΛfH

d − fdΛaHs + fdΛfdH}. (3.23)

This approach, for the sidelobe shaping of a UCA pattern, was proposed in [54]. In that case, the steering vector was obtained by applying a numerical optimisation approach similar to the one presented in Sec.3.2.2. Here, a closed form expression for the optimal beamformer is retrieved.

As a function of the steering vector s, which is the only unknown in the second term of (3.22), the desired minimisation argument is obtained by imposing the equality to

Antenna Array Signal Processing for Multistatic Radar Systems

Antenna Array Signal Processing for

Multistatic Radar Systems

Antenna Array Signal Processing for

Multistatic Radar Systems

Proefschrift

Francesco BELFIORI

Contents

List of Figures

List of Tables

Chapter 1

Introduction

1.1

Passive Coherent Locator (PCL) systems

1.2

The Multiple-Input Multiple-Output radar concept

1.3

Outline of the Thesis

Chapter 2

Antenna theory and array

pattern synthesis

2.1

Main antenna parameters

2.1.1

Directivity

2.1.2

Efficiency

2.1.3

Gain

2.2

Antenna array pattern synthesis

2.2.1

Linear array pattern synthesis

2.2.2

Circular array pattern synthesis

R

2.3

Pattern synthesis in non ideal arrays

2.3.1

Mutual coupling

2.3.2

Illumination errors

2.4

Summary

Chapter 3

Digital beamforming for PCL

3.1

Digital processing scheme

3.2

Mutual coupling compensation

3.2.1

Analytical description

3.2.2

Optimisation approach for the C matrix evaluation

3.3

DBF for circular arrays

3.3.1

Phase modes theory

3.3.2

Proposed algorithm